Continuous-time Reinforcement Learning with Risk-sensitive Objective via Quadratic Variation Penalty

Core Concepts

The paper studies continuous-time reinforcement learning (RL) with a risk-sensitive objective function in the exponential form. It shows that the risk-sensitive RL problem can be transformed into an ordinary, non-risk-sensitive RL problem augmented with a quadratic variation (QV) penalty term on the value function, which captures the variability of the value-to-go along the trajectory. This characterization allows for the straightforward adaptation of existing RL algorithms to incorporate risk sensitivity by adding the realized variance of the value process.

Abstract

The paper studies continuous-time reinforcement learning (RL) with a risk-sensitive objective function in the exponential form. The risk-sensitive objective arises either as the agent's risk attitude or as a distributionally robust approach against model uncertainty.
The key contributions are:

Establishment of the q-learning theory for continuous-time risk-sensitive RL problems. The risk-sensitive q-function is shown to be characterized by a martingale condition that only differs from the non-risk-sensitive counterpart by an extra QV penalty term. This allows for the adaptation of existing RL algorithms to incorporate risk sensitivity.

Analysis of the proposed risk-sensitive q-learning algorithm for Merton's investment problem. The paper investigates the role of the temperature parameter in the entropy-regularized RL and provides convergence guarantees.

Demonstration that the conventional policy gradient representation is inadequate for risk-sensitive problems due to the nonlinear nature of the QV penalty, while q-learning offers a solution and extends to infinite horizon settings.

Numerical experiments showing that risk-sensitive RL improves the finite-sample performance in the linear-quadratic control problem compared to non-risk-sensitive approaches.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical development of risk-sensitive continuous-time reinforcement learning.

Quotes

"The risk-sensitive objective function in the exponential form is well-known to be closely related to the robustness within a family of distributions measured by the Kullback–Leibler (KL) divergence, which is also known as the robust control problems."
"The primary contribution of this paper lies in the establishment of the q-learning theory for the continuous-time risk-sensitive RL problems."
"I highlight that the virtue of entropy regularization lies on the algorithmic aspect via boosting the estimation accuracy of the optimal policy."

Key Insights Distilled From

Continuous-time Risk-sensitive Reinforcement Learning via Quadratic Variation Penalty

by Yanwei Jia at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12598.pdf

Deeper Inquiries

How can the risk sensitivity coefficient be determined endogenously in a data-driven manner, rather than being specified exogenously?

In a data-driven approach, the risk sensitivity coefficient can be determined through iterative optimization methods that aim to find the optimal value for the coefficient based on the observed data and the performance of the RL algorithm. One common method is to use reinforcement learning techniques to learn the risk sensitivity coefficient along with the policy. This can be achieved by incorporating the risk sensitivity coefficient as a parameter in the policy and updating it during the learning process.
One approach is to frame the determination of the risk sensitivity coefficient as a hyperparameter optimization problem. By treating the risk sensitivity coefficient as a hyperparameter, techniques such as grid search, random search, or more advanced optimization algorithms like Bayesian optimization or evolutionary strategies can be employed to find the optimal value that maximizes the performance of the RL algorithm.
Another method is to use model-based reinforcement learning techniques to estimate the risk sensitivity coefficient. By building a model of the environment and simulating different scenarios with varying risk sensitivity coefficients, the algorithm can learn the impact of different coefficients on the performance and adjust accordingly.
Overall, by integrating the determination of the risk sensitivity coefficient into the learning process of the RL algorithm, it can be optimized endogenously based on the data and the task at hand.

What are the potential limitations or drawbacks of the risk-sensitive RL approach compared to other risk-aware objectives, such as CVaR or certainty-equivalency?

While risk-sensitive RL offers a unique perspective on handling uncertainty and risk in reinforcement learning problems, it also comes with certain limitations and drawbacks compared to other risk-aware objectives like Conditional Value at Risk (CVaR) or certainty-equivalency.
One limitation of risk-sensitive RL is the complexity of the exponential form objective function, which may lead to challenges in optimization and convergence. The exponential form introduces non-linearity and can make the learning process more computationally intensive compared to other risk-aware objectives.
Another drawback is the sensitivity of the risk sensitivity coefficient in risk-sensitive RL. The optimal value of the coefficient can significantly impact the performance of the algorithm, and determining this value accurately can be challenging. In contrast, other risk-aware objectives like CVaR may offer more straightforward interpretations and parameterizations.
Additionally, risk-sensitive RL may struggle with scalability and generalization to complex environments. The exponential form objective function may not always capture the full spectrum of risk preferences and may not be easily adaptable to different tasks and domains.
Overall, while risk-sensitive RL has its advantages in capturing risk attitudes and uncertainties, it is essential to consider these limitations when choosing between different risk-aware objectives for reinforcement learning tasks.

Can the insights and techniques developed in this paper be extended to other types of risk-sensitive objectives beyond the exponential form?

The insights and techniques developed in the paper, such as the martingale characterization of the optimal q-function and the risk-sensitive q-learning algorithm, can be extended to other types of risk-sensitive objectives beyond the exponential form. By understanding the fundamental principles of risk-sensitive reinforcement learning and the role of risk sensitivity coefficients, these techniques can be adapted to different objective functions that capture various risk preferences.
For objectives like CVaR or certainty-equivalency, the martingale perspective and q-learning algorithms can still be applied by modifying the loss functions and constraints to align with the specific form of the risk-sensitive objective. The key lies in formulating the objective function and constraints in a way that reflects the desired risk-awareness criteria while maintaining the principles of reinforcement learning.
Furthermore, the concept of entropy regularization and exploration can be generalized to different risk-sensitive objectives to encourage learning policies that balance between exploration and exploitation under uncertainty. By adapting the algorithms and methodologies developed in this paper, researchers can explore a wide range of risk-sensitive objectives and enhance the applicability of risk-sensitive reinforcement learning in various domains.